References:

1. https://math.stackexchange.com/questions/1838617/dividing-an-equilateral-triangle-into-n-equal-possibly-non-connected-parts

2. On congruent partitions of planar regions

3. https://research.ibm.com/haifa/ponderthis/challenges/December2003.html

Question 1: Given a number N, can we construct a convex planar region that can be cut into N mutually congruent, connected, convex pieces but not into any other number of connected, mutually congruent convex pieces?

Partial Answer (guess): For prime N, there seems to be a simple way. Take a regular N-gon and mark from it N mutually congruent quadrilaterals by drawing lines from center to mid points of the N faces. Now in each quadrilateral, replace the two 'outward' edges by copies of a polyline with say p edges and with angles that are irrational fractions of pi (see ref 3 for some justification for 'irrational') in such a way that the N-gon becomes a convex Np-gon. This Np-gon seems to allow partition into N and only N pieces that are mutually congruent, convex and connected.

Remark: I have no answer for N non-prime even when the pieces are allowed to be non-convex.

Question 2: Are there convex planar regions that allow partition into mutually congruent and connected pieces only when the number of pieces is one of exactly 2 specified values - for example is there a convex region that can only be cut into 3 connected congruent pieces or 5 congruent pieces but not into any other number of congruent pieces?

Remark: Answer to question 1 can be slightly modified to yield planar regions that seem to allow partition into only N mutually congruent pieces or kN mutually congruent pieces where N and k are primes.